Midgap Defect Modes in Dielectric and Acoustic Media

We consider three dimensional lossless periodic dielectric (photonic crystals) and acoustic media having a gap in the spectrum. If such a periodic medium is perturbed by a strong enough defect, defect eigenmodes arise, localized exponentially around the defect, with the corresponding eigenvalues in the gap. We use a modified Birman-Schwinger method to derive equations for these eigenmodes and corresponding eigenvalues in the gap, in terms of the spectral attributes of an auxiliary Hilbert-Schmidt operator. We prove that in three dimensions, under some natural conditions on the periodic background, the number of eigenvalues generated in a gap of the periodic operator is finite, and give an estimate on the number of these midgap eigenvalues. In particular, we show that if the defect is weak there are no midgap eigenvalues.